Date

What we discussed/How we spent our time

Jan 14

Syllabus. Policies. Text.
The main goals of the course are defined to be:
(1) To learn what it means to say
``Mathematics is constructed to be well founded.'' To learn
which concepts and assertions depend on which others.
To learn what are the most primitive
concepts ( = set, $\in$) and the most primitive
assertions ( = axioms of set theory).
(2) To learn how to unravel the definitions of
``function'', ``number'', and ``finite'',
through layers of more and more primitive
concepts, back to ``set'' and ``$\in$''.
(3) To learn the meanings of ``truth'' and ``provability''.
To learn proof strategies.
(4) To learn formulas for counting.
Axioms of set theory.
I will occasionally post notes for Math 2001
in the form of flash cards on Quizlet. To join
our quizlet class, go
https://quizlet.com/join/mExWGGZqj.
(Test yourself on the Axioms of Set Theory with this
Quizlet link:
https://quizlet.com/_61ko6h.)

Jan 16

We began discussing naive set theory alongside axiomatic set theory.
We discussed the Axiom of Extensionality and the Axiom of the Empty Set.
We introduced Venn diagrams, the directed graph representation of the
universe of sets, and the concept of the successor of a set.
(Some Venn diagrams from the web:
1,
2,
3,
4.
The second one is not mathematically correct.)

Jan 18

We discussed the Axioms of Infinity, Pairing, and Union.
We defined the natural numbers to be the intersection
of all inductive sets.

Jan 23

First we defined subset and power set,
and introduced the Axiom of Power Set.
Then we turned to a discussion of abbreviations
in mathematics. We defined the alphabet for set theory
(variables, nonlogical symbols, logical symbols, punctuation),
and how to write formal definitions for predicates.
Worksheet 1 (+ solution sketches).

Jan 25

We discussed the relationship between
unrestricted comprehension and restricted
comprehension. We showed, through
Russell's Paradox, that the rule of
unrestricted comprehension leads to a contradiction.
We derived that there is no set of all sets.
We also explained why there is no set
containing all sets except one.

Jan 28

We wrote the proof of the theorem
``$R=\{x\;\;x\notin x\}$ is not a set''
in English. Then we discussed some of the history of the axioms of
set theory, including the introduction of the Axiom of Replacement.
(Because of the bad weather, and the fact that 30 percent of the class
was absent, the Monday quiz was made into an
ungraded practice worksheet.)

Jan 30

Read pages 2027.
We discussed the Axiom of Choice and the Axiom
of Regularity/Foundation. In the discussion
of the Axiom of Foundation we defined the notion
of an $\in$minimal element of a set.
We defined ZFC (all 10 axioms) and
ZF (all 10 axioms minus the Axiom of Choice).
We discussed classes (like the class of all sets),
explaining what classes are
and how they may differ from sets.
We showed that the Axiom of Separation
allows us to intersect any nonempty
class of sets.

Feb 1

Today we discussed De Morgan's Laws,
$\overline{X\cap Y} = \overline{X} \cup \overline{Y}$ and
$\overline{X\cup Y} = \overline{X} \cap \overline{Y}$.
We also noted that
$X\subseteq Y$ if and only if
$\overline{X}\supseteq \overline{Y}$.
Here $\overline{X}$ represents the complement of $X$
relative to some large set $A$. By referencing De Morgan's laws,
we reasoned that $\cup$ and $\cap$ satisfy ``dual'' properties.
As an example of this duality,
we proved $X\subseteq X\cup Y$ directly, and then
derived from this,
by duality, that $X\supseteq X\cap Y$.
(Quizlet allows me to write $X'$ but not $\overline{X}$,
so on Quizlet I will write De Morgan's Laws as
$(X\cap Y)' = X' \cup Y'$ and
$(X\cup Y)' = X' \cap Y'$.
Despite this duality,
we noted that there are some asymmetries between
union and intersection. The first asymmetry we noted was
that $\bigcup \emptyset$ is a valid set (it is $\emptyset$),
but $\bigcap \emptyset$ is not set.
The second asymmetry we noted about
union and intersection is that we can only
form the union of a collection that is a set, but we can
form the intersection of any collection that is a
nonempty class.
It is important that we can form these types
of intersections, since the set of natural numbers,
$\mathbb N$, is defined as the intersection of the class
of inductive sets. (We explained today why the class of
inductive sets is not a set.)
Finally, we introduced the Kuratowski encoding of ordered pairs,
namely $(a,b) := \{\{a\},\{a,b\}\}$. We stated that,
with this definition the following theorem holds:
Theorem. $(a,b)=(c,d)$ if and only if
$a=c$ and $b=d$. (Not proved yet!)
(Test yourself on set theory terminology with this
Quizlet link:
https://quizlet.com/_61ufo1.
Some of these definitions are illustrated
by examples here
https://quizlet.com/_61vmmh.)

Feb 4

We proved
Theorem. $(a,b)=(c,d)$ if and only if
$a=c$ and $b=d$.
We defined ordered triple, ordered $n$tuple, and
Cartesian product $A\times B$. We explained why,
if $A$ and $B$ are sets, then $A\times B$ is also a set.
Quiz 1.

Feb 6

Read pages 3547.
We defined relations and explained the connection
between relations and predicates.
Handout on relations.

Feb 8

Read pages 2835.
We defined functions and went over some
vocabulary for functions.
Quiz yourself!

Feb 11

Read pages 4751.
What type of mathematical object is a
domain, codomain, image, or coimage?
Any domain is a set and any set is a domain.
Any codomain is a set and any set is a codomain.
We defined the identity function on $S$, ${\rm id}_S:S\to S$,
to show how to realize any set $S$ as a domain or a codomain.
The image of a function is a subset of the codomain.
Conversely, if $B$ is any set and
$S\subseteq B$ is any subset, then
there is a function, $\iota_S:S\to B$,
the inclusion function for $S$ into $B$, for which $B$ is the codomain
and $S$ is the image.
The coimage of a function is a partition of the domain.
Conversely, if $A$ is any set and
$P$ is any partition of $A$, then
there is a function, $N_P:A\to P: a\mapsto [a]$,
the natural map for $P$, for which $A$ is the domain
and $P$ is the coimage.
Quiz 2.

Feb 13

Read pages 5165.
Directed graph representation of binary relations.
Equivalence relations are the abstraction of kernels.
Quiz yourself!

Feb 15

What is a function?
Recursion and induction, I.
Exploiting the defining property of
$\mathbb N$.
Why is recursion a valid way to define a function?
Why is induction a valid form of proof?
SMBC on induction.

Feb 18

Recursion and induction, II.
Proving the laws of arithmetic.
(Some hints.)
Quiz 3.

Feb 20

More induction proofs.
Midterm Review Sheet.

Feb 22

Cardinality, I. Finite versus infinite.
Countable versus uncountable. Ordinal numbers.
Cardinal numbers. Equipotence.
Definitions of $A=B$, $A\leqB$, $A<B$.
CantorSchroederBernstein Theorem.

Feb 25

Cardinality notes.
Proof of CantorSchroederBernstein Theorem.
Proof of Cantor's Theorem.
$\mathbb N<{\mathcal P}(\mathbb N)=\mathbb R=\mathbb R^n$.
Quiz 4.
Quiz yourself on
ordinals and cardinals!

Feb 27

Review for the midterm!.

Mar 1

Midterm!.

Mar 4

Read Subsections 4.1.14.1.2.
First logic handout.
Structures. Alphabet of symbols.
Ingredients in a compound predicate.
No quiz!
Arnie has no respect for those who have no respect for logic.

Mar 6

Read Subsection 4.1.3.
Second logic handout.
We gave a recursive definition of the set of
terms (or algebraic expressions) in some language.
Then we discussed how to give meaning
(= assign tables) to terms.

Mar 8

Read Section 3.1.
Third logic handout.
Tables for the logical connectives.
Tautology and contradiction.
Logical equivalence of propositions.
Quiz yourself on
propositional logic!

Mar 11

Fourth logic handout.
Deciding the truth of a quantified statement.
Quiz 5.

Mar 13

Snow day! Campus closed!

Mar 15

Fifth logic handout.
We proved that every truth function
equals a truth function written in disjunctive normal form.
We defined ``complete set of connectives'',
and listed some complete sets.

Mar 18

Sixth logic handout.
We defined atomic formulas and general formulas.
We discussed how to ``standardize the variables apart'',
and then how to put a formula in prenex form.
Restricted quantifiers.
Quiz 6.

Mar 20

Practice with quantifiers!

Mar 22

What is a proof?
Axioms. Laws of Deduction, including
Modus Ponens. Propositional tautologies versus
logical tautologies versus logically valid sentences.
Direct proof, proof of the contrapositive, and
proof by contradiction. Proof by cases.

Apr 1

Formal versus informal proof.
We discussed the difference between
semantic consequence ($\Sigma\models S$)
and syntactic consequence ($\Sigma\vdash S$).
Quiz 7.

Apr 3

We discussed that the goal of any proof calculus
is to be sound, complete, and decidable.
We stated the ChurchTuring Theorem about the undecidability
of the set of logical validities,
and Godel's Completeness Theorem about the
the existence of a sound, complete, decidable
proof calculus.
We discussed the use of truth tables for
designing proof strategies.

Apr 5

Read Section 6.1.
We proved the additive and multiplicative
counting principles, using induction.

Apr 8

We discussed counting independent events.
We showed that the number of functions
from $A$ to $B$ is $B^{A}$.
We gave two proofs that ${\mathcal P}(m)=2^m$.
We defined combinatorial proof.
Quiz 8.

Apr 10

We discussed this
handout
on distributions. We introduced the notations
$P(n,k) = (n)_k = n^{\underline{k}}$
for $n!/(nk)!$.

Apr 12

Read Section 6.2.
We defined the numbers $C(n,k) = {n \choose k}$
combinatorially.
We showed that these numbers may be defined recursively.
We showed that these are the numbers that arise in Pascal's Triangle.
We showed that (i) the sum of the numbers in the $n$th
row of Pascal's triangle is $2^n$,
(ii) the alternating sum of the numbers in the $n$th
row of Pascal's triangle is $0$ if $n>0$, (iii) Pascal's Triangle
is symmetric about its main diagonal, and
(iv) we indicated how to prove that
the $n$th row is a unimodal sequence.

Apr 15

Read Section 6.5.
We gave two proofs of the
binomial theorem.
Then we defined trinomial coefficients,
gave a formula for them, explained the recursion for them
(Pascal's Pyramid),
and stated the trinomial theorem.
We indicated how to generalize to
multinomial coefficients.
Quiz 9.

Apr 17

Read Section 6.6.
Selections with repetitions.
We introduced multisets and multichoose coefficients:
$MC(n,k) = \left({n \choose k}\right)={{n+k1}\choose k}$.

Apr 19

Read Section 6.3.
Inclusionexclusion.
Counting surjective functions.

Apr 22

Stirling numbers of the second kind. $S(n,k)$ counts the number of
partitions of an $n$element set into $k$cells.
Quiz 10.
Quiz yourself on
counting formulas!

Apr 24

More on Stirling
numbers of the second kind!
We compared results on $C(n,k)$ with the
corresponding results on $S(n,k)$.

Apr 26

Review sheet for the final!
Practice problems!
(Hints!)

Apr 29

Practice problems!
(Some hints!)
Quiz 11.

May 1

Review for the final!
